-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/
-- | Polynomials
--
-- Polynomials backed by Vector.
@package poly
@version 0.5.0.0
-- | Dense polynomials and a Num-based interface.
module Data.Poly
-- | Polynomials of one variable with coefficients from a, backed
-- by a Vector v (boxed, unboxed, storable, etc.).
--
-- Use pattern X for construction:
--
--
-- >>> (X + 1) + (X - 1) :: VPoly Integer
-- 2 * X + 0
--
-- >>> (X + 1) * (X - 1) :: UPoly Int
-- 1 * X^2 + 0 * X + (-1)
--
--
-- Polynomials are stored normalized, without leading zero coefficients,
-- so 0 * X + 1 equals to 1.
--
-- Ord instance does not make much sense mathematically, it is
-- defined only for the sake of Set, Map, etc.
data Poly (v :: Type -> Type) (a :: Type)
-- | Polynomials backed by boxed vectors.
type VPoly = Poly Vector
-- | Polynomials backed by unboxed vectors.
type UPoly = Poly Vector
-- | Convert Poly to a vector of coefficients (first element
-- corresponds to a constant term).
unPoly :: Poly v a -> v a
-- | Return a leading power and coefficient of a non-zero polynomial.
--
--
-- >>> leading ((2 * X + 1) * (2 * X^2 - 1) :: UPoly Int)
-- Just (3,4)
--
-- >>> leading (0 :: UPoly Int)
-- Nothing
--
leading :: Vector v a => Poly v a -> Maybe (Word, a)
-- | Make Poly from a list of coefficients (first element
-- corresponds to a constant term).
--
--
-- >>> :set -XOverloadedLists
--
-- >>> toPoly [1,2,3] :: VPoly Integer
-- 3 * X^2 + 2 * X + 1
--
-- >>> toPoly [0,0,0] :: UPoly Int
-- 0
--
toPoly :: (Eq a, Num a, Vector v a) => v a -> Poly v a
-- | Create a monomial from a power and a coefficient.
monomial :: (Eq a, Num a, Vector v a) => Word -> a -> Poly v a
-- | Multiply a polynomial by a monomial, expressed as a power and a
-- coefficient.
--
--
-- >>> scale 2 3 (X^2 + 1) :: UPoly Int
-- 3 * X^4 + 0 * X^3 + 3 * X^2 + 0 * X + 0
--
scale :: (Eq a, Num a, Vector v a) => Word -> a -> Poly v a -> Poly v a
-- | Create an identity polynomial.
pattern X :: (Eq a, Num a, Vector v a) => Poly v a
-- | Evaluate at a given point.
--
--
-- >>> eval (X^2 + 1 :: UPoly Int) 3
-- 10
--
eval :: (Num a, Vector v a) => Poly v a -> a -> a
-- | Substitute another polynomial instead of X.
--
--
-- >>> subst (X^2 + 1 :: UPoly Int) (X + 1 :: UPoly Int)
-- 1 * X^2 + 2 * X + 2
--
subst :: (Eq a, Num a, Vector v a, Vector w a) => Poly v a -> Poly w a -> Poly w a
-- | Take a derivative.
--
--
-- >>> deriv (X^3 + 3 * X) :: UPoly Int
-- 3 * X^2 + 0 * X + 3
--
deriv :: (Eq a, Num a, Vector v a) => Poly v a -> Poly v a
-- | Compute an indefinite integral of a polynomial, setting constant term
-- to zero.
--
--
-- >>> integral (3 * X^2 + 3) :: UPoly Double
-- 1.0 * X^3 + 0.0 * X^2 + 3.0 * X + 0.0
--
integral :: (Eq a, Fractional a, Vector v a) => Poly v a -> Poly v a
-- | Polynomial division with remainder.
--
--
-- >>> quotRemFractional (X^3 + 2) (X^2 - 1 :: UPoly Double)
-- (1.0 * X + 0.0,1.0 * X + 2.0)
--
quotRemFractional :: (Eq a, Fractional a, Vector v a) => Poly v a -> Poly v a -> (Poly v a, Poly v a)
-- | Convert from dense to sparse polynomials.
--
--
-- >>> :set -XFlexibleContexts
--
-- >>> denseToSparse (1 + Data.Poly.X^2) :: Data.Poly.Sparse.UPoly Int
-- 1 * X^2 + 1
--
denseToSparse :: (Eq a, Num a, Vector v a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Convert from sparse to dense polynomials.
--
--
-- >>> :set -XFlexibleContexts
--
-- >>> sparseToDense (1 + Data.Poly.Sparse.X^2) :: Data.Poly.UPoly Int
-- 1 * X^2 + 0 * X + 1
--
sparseToDense :: (Num a, Vector v a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Laurent polynomials.
module Data.Poly.Laurent
-- | Laurent polynomials of one variable with coefficients from
-- a, backed by a Vector v (boxed, unboxed,
-- storable, etc.).
--
-- Use pattern X and operator ^- for construction:
--
--
-- >>> (X + 1) + (X^-1 - 1) :: VLaurent Integer
-- 1 * X + 0 + 1 * X^-1
--
-- >>> (X + 1) * (1 - X^-1) :: ULaurent Int
-- 1 * X + 0 + (-1) * X^-1
--
--
-- Polynomials are stored normalized, without leading and trailing zero
-- coefficients, so 0 * X + 1 + 0 * X^-1 equals to 1.
--
-- Ord instance does not make much sense mathematically, it is
-- defined only for the sake of Set, Map, etc.
data Laurent (v :: Type -> Type) (a :: Type)
-- | Laurent polynomials backed by boxed vectors.
type VLaurent = Laurent Vector
-- | Laurent polynomials backed by unboxed vectors.
type ULaurent = Laurent Vector
-- | Deconstruct a Laurent polynomial into an offset (largest
-- possible) and a regular polynomial.
--
--
-- >>> unLaurent (2 * X + 1 :: ULaurent Int)
-- (0,2 * X + 1)
--
-- >>> unLaurent (1 + 2 * X^-1 :: ULaurent Int)
-- (-1,1 * X + 2)
--
-- >>> unLaurent (2 * X^2 + X :: ULaurent Int)
-- (1,2 * X + 1)
--
-- >>> unLaurent (0 :: ULaurent Int)
-- (0,0)
--
unLaurent :: Laurent v a -> (Int, Poly v a)
-- | Construct Laurent polynomial from an offset and a regular
-- polynomial. One can imagine it as scale, but allowing negative
-- offsets.
--
--
-- >>> toLaurent 2 (2 * Data.Poly.X + 1) :: ULaurent Int
-- 2 * X^3 + 1 * X^2
--
-- >>> toLaurent (-2) (2 * Data.Poly.X + 1) :: ULaurent Int
-- 2 * X^-1 + 1 * X^-2
--
toLaurent :: (Eq a, Semiring a, Vector v a) => Int -> Poly v a -> Laurent v a
-- | Return a leading power and coefficient of a non-zero polynomial.
--
--
-- >>> leading ((2 * X + 1) * (2 * X^2 - 1) :: ULaurent Int)
-- Just (3,4)
--
-- >>> leading (0 :: ULaurent Int)
-- Nothing
--
leading :: Vector v a => Laurent v a -> Maybe (Int, a)
-- | Create a monomial from a power and a coefficient.
monomial :: (Eq a, Semiring a, Vector v a) => Int -> a -> Laurent v a
-- | Multiply a polynomial by a monomial, expressed as a power and a
-- coefficient.
--
--
-- >>> scale 2 3 (X^-2 + 1) :: ULaurent Int
-- 3 * X^2 + 0 * X + 3
--
scale :: (Eq a, Semiring a, Vector v a) => Int -> a -> Laurent v a -> Laurent v a
-- | Create an identity polynomial.
pattern X :: (Eq a, Semiring a, Vector v a) => Laurent v a
-- | This operator can be applied only to monomials with unit coefficients,
-- but is instrumental to express Laurent polynomials in mathematical
-- fashion:
--
--
-- >>> X + 2 + 3 * (X^2)^-1 :: ULaurent Int
-- 1 * X + 2 + 0 * X^-1 + 3 * X^-2
--
(^-) :: (Eq a, Num a, Vector v a) => Laurent v a -> Int -> Laurent v a
-- | Evaluate at a given point.
--
--
-- >>> eval (X^-2 + 1 :: ULaurent Double) 2
-- 1.25
--
eval :: (Field a, Vector v a) => Laurent v a -> a -> a
-- | Substitute another polynomial instead of X.
--
--
-- >>> import Data.Poly (UPoly)
--
-- >>> subst (Data.Poly.X^2 + 1 :: UPoly Int) (X^-1 + 1 :: ULaurent Int)
-- 2 + 2 * X^-1 + 1 * X^-2
--
subst :: (Eq a, Semiring a, Vector v a, Vector w a) => Poly v a -> Laurent w a -> Laurent w a
-- | Take a derivative.
--
--
-- >>> deriv (X^-1 + 3 * X) :: ULaurent Int
-- 3 + 0 * X^-1 + (-1) * X^-2
--
deriv :: (Eq a, Ring a, Vector v a) => Laurent v a -> Laurent v a
-- | Sparse multivariate polynomials with Num instance.
module Data.Poly.Multi
-- | Sparse polynomials of n variables with coefficients from
-- a, backed by a Vector v (boxed, unboxed,
-- storable, etc.).
--
-- Use patterns X, Y and Z for construction:
--
--
-- >>> :set -XDataKinds
--
-- >>> (X + 1) + (Y - 1) + Z :: VMultiPoly 3 Integer
-- 1 * X + 1 * Y + 1 * Z
--
-- >>> (X + 1) * (Y - 1) :: UMultiPoly 2 Int
-- 1 * X * Y + (-1) * X + 1 * Y + (-1)
--
--
-- Polynomials are stored normalized, without zero coefficients, so 0 *
-- X + 1 equals to 1.
--
-- Ord instance does not make much sense mathematically, it is
-- defined only for the sake of Set, Map, etc.
data MultiPoly (v :: Type -> Type) (n :: Nat) (a :: Type)
-- | Multivariate polynomials backed by boxed vectors.
type VMultiPoly (n :: Nat) (a :: Type) = MultiPoly Vector n a
-- | Multivariate polynomials backed by unboxed vectors.
type UMultiPoly (n :: Nat) (a :: Type) = MultiPoly Vector n a
-- | Convert MultiPoly to a vector of (powers, coefficient) pairs.
unMultiPoly :: MultiPoly v n a -> v (Vector n Word, a)
-- | Make MultiPoly from a list of (powers, coefficient) pairs.
--
--
-- >>> :set -XOverloadedLists -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> toMultiPoly [(fromTuple (0,0),1),(fromTuple (0,1),2),(fromTuple (1,0),3)] :: VMultiPoly 2 Integer
-- 3 * X + 2 * Y + 1
--
-- >>> toMultiPoly [(fromTuple (0,0),0),(fromTuple (0,1),0),(fromTuple (1,0),0)] :: UMultiPoly 2 Int
-- 0
--
toMultiPoly :: (Eq a, Num a, Vector v (Vector n Word, a)) => v (Vector n Word, a) -> MultiPoly v n a
-- | Create a monomial from powers and a coefficient.
monomial :: (Eq a, Num a, Vector v (Vector n Word, a)) => Vector n Word -> a -> MultiPoly v n a
-- | Multiply a polynomial by a monomial, expressed as powers and a
-- coefficient.
--
--
-- >>> :set -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> scale (fromTuple (1, 1)) 3 (X^2 + Y) :: UMultiPoly 2 Int
-- 3 * X^3 * Y + 3 * X * Y^2
--
scale :: (Eq a, Num a, KnownNat n, Vector v (Vector n Word, a)) => Vector n Word -> a -> MultiPoly v n a -> MultiPoly v n a
-- | Create a polynomial equal to the first variable.
pattern X :: (Eq a, Num a, KnownNat n, 1 <= n, Vector v (Vector n Word, a)) => MultiPoly v n a
-- | Create a polynomial equal to the second variable.
pattern Y :: (Eq a, Num a, KnownNat n, 2 <= n, Vector v (Vector n Word, a)) => MultiPoly v n a
-- | Create a polynomial equal to the third variable.
pattern Z :: (Eq a, Num a, KnownNat n, 3 <= n, Vector v (Vector n Word, a)) => MultiPoly v n a
-- | Evaluate at a given point.
--
--
-- >>> :set -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> eval (X^2 + Y^2 :: UMultiPoly 2 Int) (fromTuple (3, 4) :: Data.Vector.Sized.Vector 2 Int)
-- 25
--
eval :: (Num a, Vector v (Vector n Word, a), Vector u a) => MultiPoly v n a -> Vector u n a -> a
-- | Substitute another polynomials instead of variables.
--
--
-- >>> :set -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> subst (X^2 + Y^2 + Z^2 :: UMultiPoly 3 Int) (fromTuple (X + 1, Y + 1, X + Y :: UMultiPoly 2 Int))
-- 2 * X^2 + 2 * X * Y + 2 * X + 2 * Y^2 + 2 * Y + 2
--
subst :: (Eq a, Num a, KnownNat m, Vector v (Vector n Word, a), Vector w (Vector m Word, a)) => MultiPoly v n a -> Vector n (MultiPoly w m a) -> MultiPoly w m a
-- | Take a derivative with respect to the i-th variable.
--
--
-- >>> :set -XDataKinds
--
-- >>> deriv 0 (X^3 + 3 * Y) :: UMultiPoly 2 Int
-- 3 * X^2
--
-- >>> deriv 1 (X^3 + 3 * Y) :: UMultiPoly 2 Int
-- 3
--
deriv :: (Eq a, Num a, Vector v (Vector n Word, a)) => Finite n -> MultiPoly v n a -> MultiPoly v n a
-- | Compute an indefinite integral of a polynomial by the i-th
-- variable, setting constant term to zero.
--
--
-- >>> :set -XDataKinds
--
-- >>> integral 0 (3 * X^2 + 2 * Y) :: UMultiPoly 2 Double
-- 1.0 * X^3 + 2.0 * X * Y
--
-- >>> integral 1 (3 * X^2 + 2 * Y) :: UMultiPoly 2 Double
-- 3.0 * X^2 * Y + 1.0 * Y^2
--
integral :: (Fractional a, Vector v (Vector n Word, a)) => Finite n -> MultiPoly v n a -> MultiPoly v n a
-- | Interpret a multivariate polynomial over 1+m variables as a
-- univariate polynomial, whose coefficients are multivariate polynomials
-- over the last m variables.
segregate :: (Vector v (Vector (1 + m) Word, a), Vector v (Vector m Word, a)) => MultiPoly v (1 + m) a -> VPoly (MultiPoly v m a)
-- | Interpret a univariate polynomials, whose coefficients are
-- multivariate polynomials over the first m variables, as a
-- multivariate polynomial over 1+m variables.
unsegregate :: (Vector v (Vector (1 + m) Word, a), Vector v (Vector m Word, a)) => VPoly (MultiPoly v m a) -> MultiPoly v (1 + m) a
-- | Sparse multivariate Laurent polynomials.
module Data.Poly.Multi.Laurent
-- | Sparse Laurent polynomials of n variables with
-- coefficients from a, backed by a Vector v
-- (boxed, unboxed, storable, etc.).
--
-- Use patterns X, Y, Z and operator ^- for
-- construction:
--
--
-- >>> (X + 1) + (Y^-1 - 1) :: VMultiLaurent 2 Integer
-- 1 * X + 1 * Y^-1
--
-- >>> (X + 1) * (Z - X^-1) :: UMultiLaurent 3 Int
-- 1 * X * Z + 1 * Z + (-1) + (-1) * X^-1
--
--
-- Polynomials are stored normalized, without zero coefficients, so 0 * X
-- + 1 + 0 * X^-1 equals to 1.
--
-- Ord instance does not make much sense mathematically, it is
-- defined only for the sake of Set, Map, etc.
data MultiLaurent (v :: Type -> Type) (n :: Nat) (a :: Type)
-- | Multivariate Laurent polynomials backed by boxed vectors.
type VMultiLaurent (n :: Nat) (a :: Type) = MultiLaurent Vector n a
-- | Multivariate Laurent polynomials backed by unboxed vectors.
type UMultiLaurent (n :: Nat) (a :: Type) = MultiLaurent Vector n a
-- | Deconstruct a MultiLaurent polynomial into an offset (largest
-- possible) and a regular polynomial.
--
--
-- >>> unMultiLaurent (2 * X + 1 :: UMultiLaurent 2 Int)
-- (Vector [0,0],2 * X + 1)
--
-- >>> unMultiLaurent (1 + 2 * X^-1 :: UMultiLaurent 2 Int)
-- (Vector [-1,0],1 * X + 2)
--
-- >>> unMultiLaurent (2 * X^2 + X :: UMultiLaurent 2 Int)
-- (Vector [1,0],2 * X + 1)
--
-- >>> unMultiLaurent (0 :: UMultiLaurent 2 Int)
-- (Vector [0,0],0)
--
unMultiLaurent :: MultiLaurent v n a -> (Vector n Int, MultiPoly v n a)
-- | Construct MultiLaurent polynomial from an offset and a regular
-- polynomial. One can imagine it as scale, but allowing negative
-- offsets.
--
--
-- >>> :set -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> toMultiLaurent (fromTuple (2, 0)) (2 * Data.Poly.Multi.X + 1) :: UMultiLaurent 2 Int
-- 2 * X^3 + 1 * X^2
--
-- >>> toMultiLaurent (fromTuple (0, -2)) (2 * Data.Poly.Multi.X + 1) :: UMultiLaurent 2 Int
-- 2 * X * Y^-2 + 1 * Y^-2
--
toMultiLaurent :: (KnownNat n, Vector v (Vector n Word, a)) => Vector n Int -> MultiPoly v n a -> MultiLaurent v n a
-- | Create a monomial from a power and a coefficient.
monomial :: (Eq a, Semiring a, KnownNat n, Vector v (Vector n Word, a)) => Vector n Int -> a -> MultiLaurent v n a
-- | Multiply a polynomial by a monomial, expressed as a power and a
-- coefficient.
--
--
-- >>> :set -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> scale (fromTuple (1, 1)) 3 (X^-2 + Y) :: UMultiLaurent 2 Int
-- 3 * X * Y^2 + 3 * X^-1 * Y
--
scale :: (Eq a, Semiring a, KnownNat n, Vector v (Vector n Word, a)) => Vector n Int -> a -> MultiLaurent v n a -> MultiLaurent v n a
-- | Create a polynomial equal to the first variable.
pattern X :: (Eq a, Semiring a, KnownNat n, 1 <= n, Vector v (Vector n Word, a)) => MultiLaurent v n a
-- | Create a polynomial equal to the second variable.
pattern Y :: (Eq a, Semiring a, KnownNat n, 2 <= n, Vector v (Vector n Word, a)) => MultiLaurent v n a
-- | Create a polynomial equal to the third variable.
pattern Z :: (Eq a, Semiring a, KnownNat n, 3 <= n, Vector v (Vector n Word, a)) => MultiLaurent v n a
-- | This operator can be applied only to monomials with unit coefficients,
-- but is still instrumental to express Laurent polynomials in
-- mathematical fashion:
--
--
-- >>> 3 * X^-1 + 2 * (Y^2)^-2 :: UMultiLaurent 2 Int
-- 2 * Y^-4 + 3 * X^-1
--
(^-) :: (Eq a, Semiring a, KnownNat n, Vector v (Vector n Word, a)) => MultiLaurent v n a -> Int -> MultiLaurent v n a
-- | Evaluate at a given point.
--
--
-- >>> :set -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> eval (X^2 + Y^-1 :: UMultiLaurent 2 Double) (fromTuple (3, 4) :: Data.Vector.Sized.Vector 2 Double)
-- 9.25
--
eval :: (Field a, Vector v (Vector n Word, a), Vector u a) => MultiLaurent v n a -> Vector u n a -> a
-- | Substitute another polynomial instead of X.
--
--
-- >>> :set -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> import Data.Poly.Multi (UMultiPoly)
--
-- >>> subst (Data.Poly.Multi.X * Data.Poly.Multi.Y :: UMultiPoly 2 Int) (fromTuple (X + Y^-1, Y + X^-1 :: UMultiLaurent 2 Int))
-- 1 * X * Y + 2 + 1 * X^-1 * Y^-1
--
subst :: (Eq a, Semiring a, KnownNat n, Vector v (Vector n Word, a), Vector w (Vector n Word, a)) => MultiPoly v n a -> Vector n (MultiLaurent w n a) -> MultiLaurent w n a
-- | Take a derivative with respect to the i-th variable.
--
--
-- >>> :set -XDataKinds
--
-- >>> deriv 0 (X^3 + 3 * Y) :: UMultiLaurent 2 Int
-- 3 * X^2
--
-- >>> deriv 1 (X^3 + 3 * Y) :: UMultiLaurent 2 Int
-- 3
--
deriv :: (Eq a, Ring a, KnownNat n, Vector v (Vector n Word, a)) => Finite n -> MultiLaurent v n a -> MultiLaurent v n a
-- | Interpret a multivariate Laurent polynomial over 1+m variables
-- as a univariate Laurent polynomial, whose coefficients are
-- multivariate Laurent polynomials over the last m variables.
segregate :: (KnownNat m, Vector v (Vector (1 + m) Word, a), Vector v (Vector m Word, a)) => MultiLaurent v (1 + m) a -> VLaurent (MultiLaurent v m a)
-- | Interpret a univariate Laurent polynomials, whose coefficients are
-- multivariate Laurent polynomials over the first m variables, as
-- a multivariate polynomial over 1+m variables.
unsegregate :: forall v m a. (KnownNat m, KnownNat (1 + m), Vector v (Vector (1 + m) Word, a), Vector v (Vector m Word, a)) => VLaurent (MultiLaurent v m a) -> MultiLaurent v (1 + m) a
-- | Sparse multivariate polynomials with Semiring instance.
module Data.Poly.Multi.Semiring
-- | Sparse polynomials of n variables with coefficients from
-- a, backed by a Vector v (boxed, unboxed,
-- storable, etc.).
--
-- Use patterns X, Y and Z for construction:
--
--
-- >>> :set -XDataKinds
--
-- >>> (X + 1) + (Y - 1) + Z :: VMultiPoly 3 Integer
-- 1 * X + 1 * Y + 1 * Z
--
-- >>> (X + 1) * (Y - 1) :: UMultiPoly 2 Int
-- 1 * X * Y + (-1) * X + 1 * Y + (-1)
--
--
-- Polynomials are stored normalized, without zero coefficients, so 0 *
-- X + 1 equals to 1.
--
-- Ord instance does not make much sense mathematically, it is
-- defined only for the sake of Set, Map, etc.
data MultiPoly (v :: Type -> Type) (n :: Nat) (a :: Type)
-- | Multivariate polynomials backed by boxed vectors.
type VMultiPoly (n :: Nat) (a :: Type) = MultiPoly Vector n a
-- | Multivariate polynomials backed by unboxed vectors.
type UMultiPoly (n :: Nat) (a :: Type) = MultiPoly Vector n a
-- | Convert MultiPoly to a vector of (powers, coefficient) pairs.
unMultiPoly :: MultiPoly v n a -> v (Vector n Word, a)
-- | Make MultiPoly from a list of (powers, coefficient) pairs.
--
--
-- >>> :set -XOverloadedLists -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> toMultiPoly [(fromTuple (0,0),1),(fromTuple (0,1),2),(fromTuple (1,0),3)] :: VMultiPoly 2 Integer
-- 3 * X + 2 * Y + 1
--
-- >>> toMultiPoly [(fromTuple (0,0),0),(fromTuple (0,1),0),(fromTuple (1,0),0)] :: UMultiPoly 2 Int
-- 0
--
toMultiPoly :: (Eq a, Semiring a, Vector v (Vector n Word, a)) => v (Vector n Word, a) -> MultiPoly v n a
-- | Create a monomial from powers and a coefficient.
monomial :: (Eq a, Semiring a, Vector v (Vector n Word, a)) => Vector n Word -> a -> MultiPoly v n a
-- | Multiply a polynomial by a monomial, expressed as powers and a
-- coefficient.
--
--
-- >>> :set -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> scale (fromTuple (1, 1)) 3 (X^2 + Y) :: UMultiPoly 2 Int
-- 3 * X^3 * Y + 3 * X * Y^2
--
scale :: (Eq a, Semiring a, KnownNat n, Vector v (Vector n Word, a)) => Vector n Word -> a -> MultiPoly v n a -> MultiPoly v n a
-- | Create a polynomial equal to the first variable.
pattern X :: (Eq a, Semiring a, KnownNat n, 1 <= n, Vector v (Vector n Word, a)) => MultiPoly v n a
-- | Create a polynomial equal to the second variable.
pattern Y :: (Eq a, Semiring a, KnownNat n, 2 <= n, Vector v (Vector n Word, a)) => MultiPoly v n a
-- | Create a polynomial equal to the third variable.
pattern Z :: (Eq a, Semiring a, KnownNat n, 3 <= n, Vector v (Vector n Word, a)) => MultiPoly v n a
-- | Evaluate at a given point.
--
--
-- >>> :set -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> eval (X^2 + Y^2 :: UMultiPoly 2 Int) (fromTuple (3, 4) :: Data.Vector.Sized.Vector 2 Int)
-- 25
--
eval :: (Semiring a, Vector v (Vector n Word, a), Vector u a) => MultiPoly v n a -> Vector u n a -> a
-- | Substitute another polynomials instead of variables.
--
--
-- >>> :set -XDataKinds
--
-- >>> import Data.Vector.Generic.Sized (fromTuple)
--
-- >>> subst (X^2 + Y^2 + Z^2 :: UMultiPoly 3 Int) (fromTuple (X + 1, Y + 1, X + Y :: UMultiPoly 2 Int))
-- 2 * X^2 + 2 * X * Y + 2 * X + 2 * Y^2 + 2 * Y + 2
--
subst :: (Eq a, Semiring a, KnownNat m, Vector v (Vector n Word, a), Vector w (Vector m Word, a)) => MultiPoly v n a -> Vector n (MultiPoly w m a) -> MultiPoly w m a
-- | Take a derivative with respect to the i-th variable.
--
--
-- >>> :set -XDataKinds
--
-- >>> deriv 0 (X^3 + 3 * Y) :: UMultiPoly 2 Int
-- 3 * X^2
--
-- >>> deriv 1 (X^3 + 3 * Y) :: UMultiPoly 2 Int
-- 3
--
deriv :: (Eq a, Semiring a, Vector v (Vector n Word, a)) => Finite n -> MultiPoly v n a -> MultiPoly v n a
-- | Compute an indefinite integral of a polynomial by the i-th
-- variable, setting constant term to zero.
--
--
-- >>> :set -XDataKinds
--
-- >>> integral 0 (3 * X^2 + 2 * Y) :: UMultiPoly 2 Double
-- 1.0 * X^3 + 2.0 * X * Y
--
-- >>> integral 1 (3 * X^2 + 2 * Y) :: UMultiPoly 2 Double
-- 3.0 * X^2 * Y + 1.0 * Y^2
--
integral :: (Field a, Vector v (Vector n Word, a)) => Finite n -> MultiPoly v n a -> MultiPoly v n a
-- | Interpret a multivariate polynomial over 1+m variables as a
-- univariate polynomial, whose coefficients are multivariate polynomials
-- over the last m variables.
segregate :: (Vector v (Vector (1 + m) Word, a), Vector v (Vector m Word, a)) => MultiPoly v (1 + m) a -> VPoly (MultiPoly v m a)
-- | Interpret a univariate polynomials, whose coefficients are
-- multivariate polynomials over the first m variables, as a
-- multivariate polynomial over 1+m variables.
unsegregate :: (Vector v (Vector (1 + m) Word, a), Vector v (Vector m Word, a)) => VPoly (MultiPoly v m a) -> MultiPoly v (1 + m) a
-- | Dense polynomials and a Semiring-based interface.
module Data.Poly.Semiring
-- | Polynomials of one variable with coefficients from a, backed
-- by a Vector v (boxed, unboxed, storable, etc.).
--
-- Use pattern X for construction:
--
--
-- >>> (X + 1) + (X - 1) :: VPoly Integer
-- 2 * X + 0
--
-- >>> (X + 1) * (X - 1) :: UPoly Int
-- 1 * X^2 + 0 * X + (-1)
--
--
-- Polynomials are stored normalized, without leading zero coefficients,
-- so 0 * X + 1 equals to 1.
--
-- Ord instance does not make much sense mathematically, it is
-- defined only for the sake of Set, Map, etc.
data Poly (v :: Type -> Type) (a :: Type)
-- | Polynomials backed by boxed vectors.
type VPoly = Poly Vector
-- | Polynomials backed by unboxed vectors.
type UPoly = Poly Vector
-- | Convert Poly to a vector of coefficients (first element
-- corresponds to a constant term).
unPoly :: Poly v a -> v a
-- | Return a leading power and coefficient of a non-zero polynomial.
--
--
-- >>> leading ((2 * X + 1) * (2 * X^2 - 1) :: UPoly Int)
-- Just (3,4)
--
-- >>> leading (0 :: UPoly Int)
-- Nothing
--
leading :: Vector v a => Poly v a -> Maybe (Word, a)
-- | Make Poly from a vector of coefficients (first element
-- corresponds to a constant term).
--
--
-- >>> :set -XOverloadedLists
--
-- >>> toPoly [1,2,3] :: VPoly Integer
-- 3 * X^2 + 2 * X + 1
--
-- >>> toPoly [0,0,0] :: UPoly Int
-- 0
--
toPoly :: (Eq a, Semiring a, Vector v a) => v a -> Poly v a
-- | Create a monomial from a power and a coefficient.
monomial :: (Eq a, Semiring a, Vector v a) => Word -> a -> Poly v a
-- | Multiply a polynomial by a monomial, expressed as a power and a
-- coefficient.
--
--
-- >>> scale 2 3 (X^2 + 1) :: UPoly Int
-- 3 * X^4 + 0 * X^3 + 3 * X^2 + 0 * X + 0
--
scale :: (Eq a, Semiring a, Vector v a) => Word -> a -> Poly v a -> Poly v a
-- | Create an identity polynomial.
pattern X :: (Eq a, Semiring a, Vector v a) => Poly v a
-- | Evaluate at a given point.
--
--
-- >>> eval (X^2 + 1 :: UPoly Int) 3
-- 10
--
eval :: (Semiring a, Vector v a) => Poly v a -> a -> a
-- | Substitute another polynomial instead of X.
--
--
-- >>> subst (X^2 + 1 :: UPoly Int) (X + 1 :: UPoly Int)
-- 1 * X^2 + 2 * X + 2
--
subst :: (Eq a, Semiring a, Vector v a, Vector w a) => Poly v a -> Poly w a -> Poly w a
-- | Take a derivative.
--
--
-- >>> deriv (X^3 + 3 * X) :: UPoly Int
-- 3 * X^2 + 0 * X + 3
--
deriv :: (Eq a, Semiring a, Vector v a) => Poly v a -> Poly v a
-- | Compute an indefinite integral of a polynomial, setting constant term
-- to zero.
--
--
-- >>> integral (3 * X^2 + 3) :: UPoly Double
-- 1.0 * X^3 + 0.0 * X^2 + 3.0 * X + 0.0
--
integral :: (Eq a, Field a, Vector v a) => Poly v a -> Poly v a
-- | Convert from dense to sparse polynomials.
--
--
-- >>> :set -XFlexibleContexts
--
-- >>> denseToSparse (1 `plus` Data.Poly.X^2) :: Data.Poly.Sparse.UPoly Int
-- 1 * X^2 + 1
--
denseToSparse :: (Eq a, Semiring a, Vector v a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Convert from sparse to dense polynomials.
--
--
-- >>> :set -XFlexibleContexts
--
-- >>> sparseToDense (1 `plus` Data.Poly.Sparse.X^2) :: Data.Poly.UPoly Int
-- 1 * X^2 + 0 * X + 1
--
sparseToDense :: (Semiring a, Vector v a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Discrete Fourier transform <math>.
dft :: (Ring a, Vector v a) => a -> v a -> v a
-- | Inverse discrete Fourier transform <math>.
inverseDft :: (Field a, Vector v a) => a -> v a -> v a
-- | Multiplication of polynomials using discrete Fourier transform.
-- It could be faster than '(*)' for large polynomials if multiplication
-- of coefficients is particularly expensive.
dftMult :: (Eq a, Field a, Vector v a) => (Int -> a) -> Poly v a -> Poly v a -> Poly v a
-- | Classical orthogonal polynomials.
module Data.Poly.Orthogonal
-- | Legendre polynomials.
--
--
-- >>> take 3 legendre :: [Data.Poly.VPoly Double]
-- [1.0,1.0 * X + 0.0,1.5 * X^2 + 0.0 * X + (-0.5)]
--
legendre :: (Eq a, Field a, Vector v a) => [Poly v a]
-- | Shifted Legendre polynomials.
--
--
-- >>> take 3 legendreShifted :: [Data.Poly.VPoly Integer]
-- [1,2 * X + (-1),6 * X^2 + (-6) * X + 1]
--
legendreShifted :: (Eq a, Euclidean a, Ring a, Vector v a) => [Poly v a]
-- | Gegenbauer polynomials.
gegenbauer :: (Eq a, Field a, Vector v a) => a -> [Poly v a]
-- | Jacobi polynomials.
jacobi :: (Eq a, Field a, Vector v a) => a -> a -> [Poly v a]
-- | Chebyshev polynomials of the first kind.
--
--
-- >>> take 3 chebyshev1 :: [VPoly Integer]
-- [1,1 * X + 0,2 * X^2 + 0 * X + (-1)]
--
chebyshev1 :: (Eq a, Ring a, Vector v a) => [Poly v a]
-- | Chebyshev polynomials of the second kind.
--
--
-- >>> take 3 chebyshev2 :: [VPoly Integer]
-- [1,2 * X + 0,4 * X^2 + 0 * X + (-1)]
--
chebyshev2 :: (Eq a, Ring a, Vector v a) => [Poly v a]
-- | Probabilists' Hermite polynomials.
--
--
-- >>> take 3 hermiteProb :: [VPoly Integer]
-- [1,1 * X + 0,1 * X^2 + 0 * X + (-1)]
--
hermiteProb :: (Eq a, Ring a, Vector v a) => [Poly v a]
-- | Physicists' Hermite polynomials.
--
--
-- >>> take 3 hermitePhys :: [VPoly Double]
-- [1.0,2.0 * X + 0.0,4.0 * X^2 + 0.0 * X + (-2.0)]
--
hermitePhys :: (Eq a, Ring a, Vector v a) => [Poly v a]
-- | Laguerre polynomials.
--
--
-- >>> take 3 laguerre :: [VPoly Double]
-- [1.0,(-1.0) * X + 1.0,0.5 * X^2 + (-2.0) * X + 1.0]
--
laguerre :: (Eq a, Field a, Vector v a) => [Poly v a]
-- | Generalized Laguerre polynomials
laguerreGen :: (Eq a, Field a, Vector v a) => a -> [Poly v a]
-- | Sparse polynomials with Num instance.
module Data.Poly.Sparse
-- | Sparse univariate polynomials with coefficients from a,
-- backed by a Vector v (boxed, unboxed, storable, etc.).
--
-- Use pattern X for construction:
--
--
-- >>> (X + 1) + (X - 1) :: VPoly Integer
-- 2 * X
--
-- >>> (X + 1) * (X - 1) :: UPoly Int
-- 1 * X^2 + (-1)
--
--
-- Polynomials are stored normalized, without zero coefficients, so 0 *
-- X + 1 equals to 1.
--
-- Ord instance does not make much sense mathematically, it is
-- defined only for the sake of Set, Map, etc.
type Poly (v :: Type -> Type) (a :: Type) = MultiPoly v 1 a
-- | Polynomials backed by boxed vectors.
type VPoly (a :: Type) = Poly Vector a
-- | Polynomials backed by unboxed vectors.
type UPoly (a :: Type) = Poly Vector a
-- | Convert Poly to a vector of coefficients.
unPoly :: (Vector v (Word, a), Vector v (Vector 1 Word, a)) => Poly v a -> v (Word, a)
-- | Make Poly from a list of (power, coefficient) pairs.
--
--
-- >>> :set -XOverloadedLists
--
-- >>> toPoly [(0,1),(1,2),(2,3)] :: VPoly Integer
-- 3 * X^2 + 2 * X + 1
--
-- >>> toPoly [(0,0),(1,0),(2,0)] :: UPoly Int
-- 0
--
toPoly :: (Eq a, Num a, Vector v (Word, a), Vector v (Vector 1 Word, a)) => v (Word, a) -> Poly v a
-- | Return a leading power and coefficient of a non-zero polynomial.
--
--
-- >>> import Data.Poly.Sparse (UPoly)
--
-- >>> leading ((2 * X + 1) * (2 * X^2 - 1) :: UPoly Int)
-- Just (3,4)
--
-- >>> leading (0 :: UPoly Int)
-- Nothing
--
leading :: Vector v (Vector 1 Word, a) => Poly v a -> Maybe (Word, a)
-- | Create a monomial from a power and a coefficient.
monomial :: (Eq a, Num a, Vector v (Vector 1 Word, a)) => Word -> a -> Poly v a
-- | Multiply a polynomial by a monomial, expressed as a power and a
-- coefficient.
--
--
-- >>> scale 2 3 (X^2 + 1) :: UPoly Int
-- 3 * X^4 + 3 * X^2
--
scale :: (Eq a, Num a, Vector v (Vector 1 Word, a)) => Word -> a -> Poly v a -> Poly v a
-- | Create an identity polynomial.
pattern X :: (Eq a, Num a, Vector v (Vector 1 Word, a)) => Poly v a
-- | Evaluate at a given point.
--
--
-- >>> eval (X^2 + 1 :: UPoly Int) 3
-- 10
--
eval :: (Num a, Vector v (Vector 1 Word, a)) => Poly v a -> a -> a
-- | Substitute another polynomial instead of X.
--
--
-- >>> subst (X^2 + 1 :: UPoly Int) (X + 1 :: UPoly Int)
-- 1 * X^2 + 2 * X + 2
--
subst :: (Eq a, Num a, Vector v (Vector 1 Word, a), Vector w (Vector 1 Word, a)) => Poly v a -> Poly w a -> Poly w a
-- | Take a derivative.
--
--
-- >>> deriv (X^3 + 3 * X) :: UPoly Int
-- 3 * X^2 + 3
--
deriv :: (Eq a, Num a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Compute an indefinite integral of a polynomial, setting constant term
-- to zero.
--
--
-- >>> integral (3 * X^2 + 3) :: UPoly Double
-- 1.0 * X^3 + 3.0 * X
--
integral :: (Fractional a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Polynomial division with remainder.
--
--
-- >>> quotRemFractional (X^3 + 2) (X^2 - 1 :: UPoly Double)
-- (1.0 * X,1.0 * X + 2.0)
--
quotRemFractional :: (Eq a, Fractional a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a -> (Poly v a, Poly v a)
-- | Convert from dense to sparse polynomials.
--
--
-- >>> :set -XFlexibleContexts
--
-- >>> denseToSparse (1 + Data.Poly.X^2) :: Data.Poly.Sparse.UPoly Int
-- 1 * X^2 + 1
--
denseToSparse :: (Eq a, Num a, Vector v a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Convert from sparse to dense polynomials.
--
--
-- >>> :set -XFlexibleContexts
--
-- >>> sparseToDense (1 + Data.Poly.Sparse.X^2) :: Data.Poly.UPoly Int
-- 1 * X^2 + 0 * X + 1
--
sparseToDense :: (Num a, Vector v a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Sparse Laurent polynomials.
module Data.Poly.Sparse.Laurent
-- | Laurent polynomials of one variable with coefficients from
-- a, backed by a Vector v (boxed, unboxed,
-- storable, etc.).
--
-- Use pattern X and operator ^- for construction:
--
--
-- >>> (X + 1) + (X^-1 - 1) :: VLaurent Integer
-- 1 * X + 1 * X^-1
--
-- >>> (X + 1) * (1 - X^-1) :: ULaurent Int
-- 1 * X + (-1) * X^-1
--
--
-- Polynomials are stored normalized, without zero coefficients, so 0 * X
-- + 1 + 0 * X^-1 equals to 1.
--
-- Ord instance does not make much sense mathematically, it is
-- defined only for the sake of Set, Map, etc.
type Laurent (v :: Type -> Type) (a :: Type) = MultiLaurent v 1 a
-- | Laurent polynomials backed by boxed vectors.
type VLaurent (a :: Type) = Laurent Vector a
-- | Laurent polynomials backed by unboxed vectors.
type ULaurent (a :: Type) = Laurent Vector a
-- | Deconstruct a Laurent polynomial into an offset (largest
-- possible) and a regular polynomial.
--
--
-- >>> unLaurent (2 * X + 1 :: ULaurent Int)
-- (0,2 * X + 1)
--
-- >>> unLaurent (1 + 2 * X^-1 :: ULaurent Int)
-- (-1,1 * X + 2)
--
-- >>> unLaurent (2 * X^2 + X :: ULaurent Int)
-- (1,2 * X + 1)
--
-- >>> unLaurent (0 :: ULaurent Int)
-- (0,0)
--
unLaurent :: Laurent v a -> (Int, Poly v a)
-- | Construct Laurent polynomial from an offset and a regular
-- polynomial. One can imagine it as scale, but allowing negative
-- offsets.
--
--
-- >>> toLaurent 2 (2 * Data.Poly.Sparse.X + 1) :: ULaurent Int
-- 2 * X^3 + 1 * X^2
--
-- >>> toLaurent (-2) (2 * Data.Poly.Sparse.X + 1) :: ULaurent Int
-- 2 * X^-1 + 1 * X^-2
--
toLaurent :: Vector v (Vector 1 Word, a) => Int -> Poly v a -> Laurent v a
-- | Return a leading power and coefficient of a non-zero polynomial.
--
--
-- >>> leading ((2 * X + 1) * (2 * X^2 - 1) :: ULaurent Int)
-- Just (3,4)
--
-- >>> leading (0 :: ULaurent Int)
-- Nothing
--
leading :: Vector v (Vector 1 Word, a) => Laurent v a -> Maybe (Int, a)
-- | Create a monomial from a power and a coefficient.
monomial :: (Eq a, Semiring a, Vector v (Vector 1 Word, a)) => Int -> a -> Laurent v a
-- | Multiply a polynomial by a monomial, expressed as a power and a
-- coefficient.
--
--
-- >>> scale 2 3 (X^-2 + 1) :: ULaurent Int
-- 3 * X^2 + 3
--
scale :: (Eq a, Semiring a, Vector v (Vector 1 Word, a)) => Int -> a -> Laurent v a -> Laurent v a
-- | Create an identity polynomial.
pattern X :: (Eq a, Semiring a, Vector v (Vector 1 Word, a)) => Laurent v a
-- | This operator can be applied only to monomials with unit coefficients,
-- but is instrumental to express Laurent polynomials in mathematical
-- fashion:
--
--
-- >>> X + 2 + 3 * (X^2)^-1 :: ULaurent Int
-- 1 * X + 2 + 3 * X^-2
--
(^-) :: (Eq a, Semiring a, Vector v (Vector 1 Word, a)) => Laurent v a -> Int -> Laurent v a
-- | Evaluate at a given point.
--
--
-- >>> eval (X^-2 + 1 :: ULaurent Double) 2
-- 1.25
--
eval :: (Field a, Vector v (Vector 1 Word, a)) => Laurent v a -> a -> a
-- | Substitute another polynomial instead of X.
--
--
-- >>> import Data.Poly.Sparse (UPoly)
--
-- >>> subst (Data.Poly.Sparse.X^2 + 1 :: UPoly Int) (X^-1 + 1 :: ULaurent Int)
-- 2 + 2 * X^-1 + 1 * X^-2
--
subst :: (Eq a, Semiring a, Vector v (Vector 1 Word, a), Vector w (Vector 1 Word, a)) => Poly v a -> Laurent w a -> Laurent w a
-- | Take a derivative.
--
--
-- >>> deriv (X^-3 + 3 * X) :: ULaurent Int
-- 3 + (-3) * X^-4
--
deriv :: (Eq a, Ring a, Vector v (Vector 1 Word, a)) => Laurent v a -> Laurent v a
-- | Sparse polynomials with Semiring instance.
module Data.Poly.Sparse.Semiring
-- | Sparse univariate polynomials with coefficients from a,
-- backed by a Vector v (boxed, unboxed, storable, etc.).
--
-- Use pattern X for construction:
--
--
-- >>> (X + 1) + (X - 1) :: VPoly Integer
-- 2 * X
--
-- >>> (X + 1) * (X - 1) :: UPoly Int
-- 1 * X^2 + (-1)
--
--
-- Polynomials are stored normalized, without zero coefficients, so 0 *
-- X + 1 equals to 1.
--
-- Ord instance does not make much sense mathematically, it is
-- defined only for the sake of Set, Map, etc.
type Poly (v :: Type -> Type) (a :: Type) = MultiPoly v 1 a
-- | Polynomials backed by boxed vectors.
type VPoly (a :: Type) = Poly Vector a
-- | Polynomials backed by unboxed vectors.
type UPoly (a :: Type) = Poly Vector a
-- | Convert Poly to a vector of coefficients.
unPoly :: (Vector v (Word, a), Vector v (Vector 1 Word, a)) => Poly v a -> v (Word, a)
-- | Make Poly from a list of (power, coefficient) pairs.
--
--
-- >>> :set -XOverloadedLists
--
-- >>> toPoly [(0,1),(1,2),(2,3)] :: VPoly Integer
-- 3 * X^2 + 2 * X + 1
--
-- >>> toPoly [(0,0),(1,0),(2,0)] :: UPoly Int
-- 0
--
toPoly :: (Eq a, Semiring a, Vector v (Word, a), Vector v (Vector 1 Word, a)) => v (Word, a) -> Poly v a
-- | Return a leading power and coefficient of a non-zero polynomial.
--
--
-- >>> import Data.Poly.Sparse (UPoly)
--
-- >>> leading ((2 * X + 1) * (2 * X^2 - 1) :: UPoly Int)
-- Just (3,4)
--
-- >>> leading (0 :: UPoly Int)
-- Nothing
--
leading :: Vector v (Vector 1 Word, a) => Poly v a -> Maybe (Word, a)
-- | Create a monomial from a power and a coefficient.
monomial :: (Eq a, Semiring a, Vector v (Vector 1 Word, a)) => Word -> a -> Poly v a
-- | Multiply a polynomial by a monomial, expressed as a power and a
-- coefficient.
--
--
-- >>> scale 2 3 (X^2 + 1) :: UPoly Int
-- 3 * X^4 + 3 * X^2
--
scale :: (Eq a, Semiring a, Vector v (Vector 1 Word, a)) => Word -> a -> Poly v a -> Poly v a
-- | Create an identity polynomial.
pattern X :: (Eq a, Semiring a, Vector v (Vector 1 Word, a)) => Poly v a
-- | Evaluate at a given point.
--
--
-- >>> eval (X^2 + 1 :: UPoly Int) 3
-- 10
--
eval :: (Semiring a, Vector v (Vector 1 Word, a)) => Poly v a -> a -> a
-- | Substitute another polynomial instead of X.
--
--
-- >>> subst (X^2 + 1 :: UPoly Int) (X + 1 :: UPoly Int)
-- 1 * X^2 + 2 * X + 2
--
subst :: (Eq a, Semiring a, Vector v (Vector 1 Word, a), Vector w (Vector 1 Word, a)) => Poly v a -> Poly w a -> Poly w a
-- | Take a derivative.
--
--
-- >>> deriv (X^3 + 3 * X) :: UPoly Int
-- 3 * X^2 + 3
--
deriv :: (Eq a, Semiring a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Compute an indefinite integral of a polynomial, setting constant term
-- to zero.
--
--
-- >>> integral (3 * X^2 + 3) :: UPoly Double
-- 1.0 * X^3 + 3.0 * X
--
integral :: (Field a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Convert from dense to sparse polynomials.
--
--
-- >>> :set -XFlexibleContexts
--
-- >>> denseToSparse (1 `plus` Data.Poly.X^2) :: Data.Poly.Sparse.UPoly Int
-- 1 * X^2 + 1
--
denseToSparse :: (Eq a, Semiring a, Vector v a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a
-- | Convert from sparse to dense polynomials.
--
--
-- >>> :set -XFlexibleContexts
--
-- >>> sparseToDense (1 `plus` Data.Poly.Sparse.X^2) :: Data.Poly.UPoly Int
-- 1 * X^2 + 0 * X + 1
--
sparseToDense :: (Semiring a, Vector v a, Vector v (Vector 1 Word, a)) => Poly v a -> Poly v a